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The purpose of this work is to generalize, in the context of 1-motives, the p-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide a local pairing between disjoint zero-cycles of degree zero on a curve, which is done by considering the Picard and Albanese 1-motives associated to the curve. [ABSTRACT FROM AUTHOR]

We show that if X n is a variety of c × n -matrices that is stable under the group Sym ⁡ ([ n ]) of column permutations and if forgetting the last column maps X n into X n - 1 , then the number of Sym ⁡ ([ n ]) -orbits on irreducible components of X n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine 퐅퐈 퐨퐩 -schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one 퐅퐈 퐨퐩 -scheme becomes of product form, where X n = Y n for some scheme Y in affine c-space. Furthermore, to any 퐅퐈 퐨퐩 -scheme of width one we associate a component functor from the category 퐅퐈 of finite sets with injections to the category 퐏퐅 of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym ⁡ ([ n ]) -orbits of components of X n , for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism. [ABSTRACT FROM AUTHOR]

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f. [ABSTRACT FROM AUTHOR]

Let ℓ ⩾ 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in ℓ variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u3 = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work. [ABSTRACT FROM AUTHOR]

We introduce and classify 1‐clustered families of linear spaces in the Grassmannian G(k−1,n)$\mathbb {G}(k-1,n)$ and give applications to Lang‐type conjectures. Let X⊂Pn$X \subset \mathbb {P}^n$ be a very general hypersurface of degree d$d$. Let ZL$Z_L$ be the locus of points contained in a line of X$X$. Let Z2$Z_2$ be the closure of the locus of points on X$X$ that are swept out by lines that meet X$X$ in at most 2 points. We prove that: if d⩾3n+22$d \geqslant \frac{3n+2}{2}$, then X$X$ is algebraically hyperbolic modulo ZL;$Z_L\hbox{\it ;}$if d⩾3n2$d \geqslant \frac{3n}{2}$, X$X$ contains lines but no other rational curves;if d⩾3n+32$d \geqslant \frac{3n+3}{2}$, then the only points on X$X$ that are rationally Chow zero equivalent to points other than themselves are contained in Z2$Z_2$;if d⩾3n+22$d \geqslant \frac{3n+2}{2}$ and a relative Green–Griffiths–Lang Conjecture holds, then the exceptional locus for X$X$ is contained in Z2$Z_2$. These sharpen prior results of Ein, Voisin, Pacienza, Clemens and Ran. [ABSTRACT FROM AUTHOR]

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F q 2 is maximal if the number N q 2 of its F q 2 -rational points attains the Hasse-Weil upper bound, that is N q 2 = q 2 + 2 g q + 1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p 2 where p is the characteristic of F q 2 . Doing so we also determine the F q 2 -isomorphism classes of such curves and describe their full F q 2 -automorphism groups. [ABSTRACT FROM AUTHOR]

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2. [ABSTRACT FROM AUTHOR]

In this paper we prove a formula for the number of rational points in a certain height box for a split toric variety over a number field. [ABSTRACT FROM AUTHOR]

In this paper, as an analog of the number field case, for an elliptic curve E defined over the algebraic numbers and for any subfield F of algebraic numbers, we say that E has the Northcott property over F if there are at most finitely many F -rational points on E of uniformly bounded height, and we say that E has the property (P) over F if for any infinite subset S of F -rational points on E , f (S) = S for an F -endomorphism f of E implies that f is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true. [ABSTRACT FROM AUTHOR]

It is shown that c = − 29 / 16 is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map f c (x) = x 2 + c has a rational periodic point of period 3. Several arithmetic conditions on the set of all such rational numbers c and the rational orbits of f c (x) are proved. A graph on the numerators of the rational 3-periodic points of maps f c is considered which reflects connections between solutions of norm equations from the cubic field of discriminant −23. [ABSTRACT FROM AUTHOR]

We complete the computation of all Q -rational points on all the 64 maximal Atkin-Lehner quotients X 0 (N) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q -rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the Q -rational points on all of their modular coverings. [ABSTRACT FROM AUTHOR]

For S ⊆ R , positive integer n, and d > 0 , let G (S n , d) be the graph whose vertex set is S n where any two vertices are adjacent if and only if they are Euclidean distance d apart. The primary question we will consider in our work is as follows. Given n and distance d actually realized as a distance between points of the rational space Q n , does there exist a finite graph G that appears as a subgraph of G (Q n , d) but not as a subgraph of G (R n - 1 , 1) ? We answer this question affirmatively for n ≤ 5 , and along the way, resolve a few related questions as well. [ABSTRACT FROM AUTHOR]

We study the Selmer varieties of smooth projective curves of genus at least two defined over |$\mathbb{Q}$| which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P |$^1$| with solvable Galois group, and in particular any superelliptic curve over |$\mathbb{Q}$|⁠ , has only finitely many rational points over |$\mathbb{Q}$|⁠. [ABSTRACT FROM AUTHOR]

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits [Ann. of Math. (2) 97 (1973), pp. 499–571]. Our method extends the approach introduced by the first author in [Proc. Lond. Math. Soc. (3) 102 (2011), pp. 951–983] to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation. [ABSTRACT FROM AUTHOR]

Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G. We determine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is generated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k). This last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups; this appears to be the first application of Ratner's theorems in the context of operator algebras. [ABSTRACT FROM AUTHOR]

In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves, and prove some results towards a matrix analogue of this problem, including a full analogue for 2×2$2\times 2$ matrices defined over C$\mathbb {C}$. [ABSTRACT FROM AUTHOR]

Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus 2 quintic curves over Q of Mordell–Weil rank 2 or 3 whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases. [ABSTRACT FROM AUTHOR]

The extended coset leader weight enumerator of the generalized Reed–Solomon [ q + 1 , q - 3 , 5 ] q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore, the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse–Weil bound on F q -rational points we show that there is a 3-plane containing the passant. [ABSTRACT FROM AUTHOR]

Given a family of quadratic twists of a fixed elliptic curve defined over Q$\mathbb {Q}$, we investigate the average rank in the subfamily of twists having a non‐torsion rational point of almost minimal height. We show in particular that the average analytic rank is greater than one. [ABSTRACT FROM AUTHOR]

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups. [ABSTRACT FROM AUTHOR]

A Diophantine m -tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1) (y 2 − 1) (z 2 − 1) = k 2 , be an affine variety over K. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x , y , z , k) ∈ X (K) is equal to k. We denote by X ‾ the projective closure of X and for a fixed k by X k a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties X k , X and X ‾ tell us about the arithmetic of Diophantine triples. First, we prove that the variety X ‾ is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface X k for a given k ∈ F p × in the prime field F p of odd characteristic and an abelian surface which is a product of two elliptic curves E k × E k where E k : y 2 = x (k 2 (1 + k 2) 3 + 2 (1 + k 2) 2 x + x 2). We derive an explicit formula for N (p , k) , the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X ‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ‾ over an arbitrary finite field F q. Using it we reprove the formula for the number of Diophantine triples over F q from [DK21]. Curiously, from the interplay of the two (K3 and rational) fibrations of X ‾ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)). Finally, in the Appendix, Luka Lasić defines circular Diophantine m -tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples. [ABSTRACT FROM AUTHOR]

For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity rc (X) . This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of X without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding rc (X) and its variant rc Q (X) , restricting the descriptions of X to rational polyhedra. As our main results we show that rc (X) = rc Q (X) when: (a) X is at most four-dimensional, (b) X represents every residue class in (Z / 2 Z) d , (c) the convex hull of X contains an interior integer point, or (d) the lattice-width of X is above a certain threshold. Additionally, rc (X) can be algorithmically computed when X is at most three-dimensional, or X satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on rc (X) in terms of the dimension of X. [ABSTRACT FROM AUTHOR]

Thus if I q i <= | I G i SB 1 sb ( I P i )| then I q i divides | I G i SB 1 sb ( I P i )| and | I G i SB 1 sb ( I P i )| is a pole number. Thus we have HT ht but this case cannot occur, see [[37], Lemma 2.31], [[7], p. 235, Section 3]. Thus we can suppose that I G i = I G i SB 0 sb ( I P i ) = I G i SB 1 sb ( I P i ) for some I P i I X i . That is g - 1 < HT ht Surprisingly enough, the situation where I i SB 1 sb = 1 or when I i SB 1 sb /= 1 and | I G i SB 1 sb | is among the two first minimal generators characterizes the Weierstrass semigroup in another way: Lemma 4 I When i SB 1 sb /= 1 I and p i SP I h i 0 sp = | I G i SB 1 sb ( I P i )| I is the first or the second generator or when i SB 1 sb = 1, I then H i ( I P i ) I is a telescopic numerical semigroup i . [Extracted from the article]

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Letting f and h denoting rational maps of degree at least 2 and 1 respectively, and denote by f n the nth iterate of f. We give upper and lower bounds on the set of accumulation points of the Julia sets J (h ∘ f n) with respect to the Hausdorff metric on non-empty compact subsets of the sphere. [ABSTRACT FROM AUTHOR]

Recently, Laohakosol and Sripayap (East-West J. Math.19 (2017) 65–79) considered certain Cantor-like power series. They investigated the nature of such series evaluated at rational points. Using Roth's theorem, they proved that these series take transcendental values at rational points subject to certain conditions on the growth of the coefficients of these series. In the present paper, we consider these Cantor-like power series treated by Laohakosol and Sripayap at algebraic points and at Liouville number points. Using a theorem due to LeVeque, a lemma due to İçen, and Laohakosol and Sripayap's technique, we first prove that these Cantor-like power series take transcendental values at algebraic points. Using Roth's theorem and Laohakosol and Sripayap's technique, we then prove that these series take rational or transcendental values at Liouville number points. [ABSTRACT FROM AUTHOR]

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques. [ABSTRACT FROM AUTHOR]

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these to almost 60,000 manifolds, providing a wealth of data about them. We use this data to explore the delicate question of counting only connected essential surfaces and propose some conjectures. Our methods involve normal and almost normal surfaces, especially the work of Tollefson and Oertel, combined with techniques pioneered by Ehrhart for counting lattice points in polyhedra with rational vertices. We also introduce a new way of testing if a normal surface in an ideal triangulation is essential that avoids cutting the manifold open along the surface; rather, we use almost normal surfaces in the original triangulation. [ABSTRACT FROM AUTHOR]

Given a subset of X ⊆ ℝn we can associate with every point x ∈ ℝn a vector space V of maximal dimension with the property that for some ball centered at x, the subset X coincides inside the ball with a union of lines parallel to V. A point is singular if V has dimension 0. In an earlier paper we proved that a 〈ℝ, +, <, ℤ〉-definable relation X is 〈ℝ, +, <, 1〉-definable if and only if the number of singular points is finite and every rational section of X is 〈R, +, <, 1〉-definable, where a rational section is a set obtained from X by fixing some component to a rational value. Here we show that we can dispense with the hypothesis of X being 〈ℝ, +, <, ℤ〉-definable by requiring that the components of the singular points be rational numbers. This provides a topological characterization of first-order definability in the structure 〈ℝ, +, <, 1〉. It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of 〈ℝ, +, <, 1〉- and 〈ℝ, +, <,ℤ〉-definability for a wide class of relations, which turns into an effective criterion provided that the corresponding theory is decidable. In particular these results apply to the class of so-called k–recognizable relations which are defined by finite Muller automata via the representation of the reals in a integer basis k, and allow us to prove that it is decidable whether a k–recognizable relation (of any arity) is l–recognizable for every base l ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we study quadratic points on the non-split Cartan modular curves X n s (p) , for p = 7 , 1 1 , and 1 3. Recently, Siksek proved that all quadratic points on X n s (7) arise as pullbacks of rational points on X n s + (7). Using similar techniques for p = 1 1 , and employing a version of Chabauty for symmetric powers of curves for p = 1 3 , we show that the same holds for X n s (1 1) and X n s (1 3). As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular. [ABSTRACT FROM AUTHOR]

In this paper, we present the Walsh transform fW: F2n → Z of some quadratic trace forms f(x) = ... over finite fields of characteristic two where the degree of extension n is even. In this article, we consider only trace forms with one or two terms where ais are coming from base field F2. We use the Walsh coefficient fW (0) to investigate the number of rational points on Artin-Schreier curves over F2n of the form χ: y² + y = .... Using these results we also derive some maximal Artin-Scheier curves. [ABSTRACT FROM AUTHOR]

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB%5fFILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also extend this polyhedrality result to mixed-integer sets defined by linear constraints. [ABSTRACT FROM AUTHOR]

We give a correct proof to the fact that all rational points on the curve y² = (x² +1)(x² +3)(x² +7) are ±∞ and (±1, ±8). This corrects previous works of Cohen [3] and Duquesne [4,5]. [ABSTRACT FROM AUTHOR]

We give an asymptotic formula for the number of rational points of bounded height of quartic hypersurfaces Q n : x 4 = (y 1 2 + ⋯ + y n 2) z 2 by following the approach and method presented by Liu, Wu, and Zhao for cubic hypersurfaces. Also, we give an observation for much higher dimension hypersurfaces of similar form. [ABSTRACT FROM AUTHOR]

Motivated by Lang–Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik's theorem and Prokhorov–Shramov's notion of quasi-minimal model but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai's finiteness result for varieties of general type and thereby generalize Noguchi's theorem (formerly Lang's conjecture). [ABSTRACT FROM AUTHOR]

A flag of codes $C_{0} \subsetneq C_{1} \subsetneq \cdots \subsetneq C_{s} \subseteq \mathbb {F}_{q} ^{n}$ is said to satisfy the isometry-dual property if there exists ${\mathbf{x}}\in (\mathbb {F}_{q}^{*})^{n}$ such that the code $C_{i}$ is x-isometric to the dual code $C_{s-i}^\perp $ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field $\mathcal {F}$ , we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $C_{\mathcal {L}}(D, a_{0}P+bQ)\subsetneq C_{\mathcal {L}}(D, a_{1}P+bQ)\subsetneq {\dots } \subsetneq C_{\mathcal {L}}(D, a_{s}P+bQ)$ , where the divisor $D$ is the sum of pairwise different rational places of $\mathcal {F}$ and $P, Q$ are not in $\mathop {\mathrm {supp}}\nolimits (D)$. We characterize those sequences in terms of $b$ for general function fields. We then apply the result to the broad class of Kummer extensions $\mathcal {F}$ defined by affine equations of the form $y^{m}=f(x)$ , for $f(x)$ a separable polynomial of degree $r$ , where $\gcd (r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$ , and for $D$ an $Aut(\mathcal {F}/ \mathbb {F}_{q})$ -invariant sum of rational places of $\mathcal {F}$ , such that $P, Q \notin \mathop {\mathrm {supp}}\nolimits D$ , it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields. [ABSTRACT FROM AUTHOR]

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be an (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form an : = f(φn(x0)), where φ : X → X and f : X → P¹ are rational maps defined over K and x0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of φ and f. We show that the set of n for which an ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero. [ABSTRACT FROM AUTHOR]

An improved asymptotic formula is established for the number of rational points of bounded height on the split smooth del Pezzo surface of degree 5. The proof uses the five conic bundle structures on the surface. [ABSTRACT FROM AUTHOR]

We consider the potential density of rational points on an algebraic variety defined over a number field K, i.e., the property that the set of rational points of X becomes Zariski dense after a finite field extension of K. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with int-amplified endomorphisms in dimension ≤3. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of classifying quadruples (K, E, m1, m2) where K is a number field, E is an elliptic curve defined over K and (m1, m2) is a pair of relatively prime positive integers for which the intersection K(E[m1]) ∩ K(E[m2]) is a non-abelian extension of K. There is an infinite set S of modular curves whose K-rational points capture all elliptic curves over K without complex multiplication that have this property. Our main theorem explicitly describes the subset S0 ⊆ S consisting of those modular curves having genus zero. The subset S0 turns out to consist of four modular curves, each isomorphic to P¹ over its field of definition. In the case K = Q, this has applications to the problem of determining when the Galois representation on the torsion of E is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example. [ABSTRACT FROM AUTHOR]

Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at general points. Résumé. Récemment, Kanemitsu a découvert un contre-exemple à la conjecture de longue date selon laquelle le faisceau tangent d’une variété de Fano de nombre de Picard un est (semi) stable. Son contreexemple est une variété horosphérique lisse. Une conjecture plus faible affirme que le fibré tangent d’une variété de Fano de nombre de Picard un est simple. Nous prouvons que cette conjecture plus faible est valable pour les variétés horosphériques lisses de nombre de Picard un. Notre preuve découle de l’existence d’une famille irréductible de courbes rationnelles non pliables dont les vecteurs tangents engendrent les espaces tangents de la variété horosphérique en des points généraux. [ABSTRACT FROM AUTHOR]

Let E be an ordinary elliptic curve over a finite field and g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E^g. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre's obstruction for principally polarized abelian threefolds isogenous to E^3 and of the Igusa modular form in dimension 4. We illustrate our algorithms with examples of curves with many rational points over finite fields. [ABSTRACT FROM AUTHOR]

Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm. In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm. [ABSTRACT FROM AUTHOR]

A positive integer N is called a θ -congruent number if there is a θ -triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N r 2 - s 2 , where θ ∈ (0 , π) , cos (θ) = s / r , and 0 ≤ | s | < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a θ -congruent number if and only if the elliptic curve E N θ : y 2 = x (x + (r + s) N) (x - (r - s) N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N ≠ 1 , 2 , 3 , 6 is a θ -congruent number if and only if rank of E N θ (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ -triangle for a θ -congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E N θ (Q) . Then, based on the addition of two distinct points in E N θ (Q) , we provide a way to find new rational θ -triangles for the θ -congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E N θ (Q) with corresponding rational θ -triangles. [ABSTRACT FROM AUTHOR]

Let K be a global field and let Z be a geometrically irreducible algebraic variety defined over K. We show that if a big set S < Z of rational points of bounded height occupies few residue classes modulo p for many prime ideals p, then a positive proportion of S must lie in the zero set of a polynomial of low degree that does not vanish at Z. This generalizes a result of Walsh who studied the case when S < {0 ... N}d. [ABSTRACT FROM AUTHOR]

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of ℙ n of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of ℚ v -soluble diagonal hypersurfaces for all places v. As worked examples, we carry out our strategy in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove that around 9 9. 9 9 % of diagonal cubic fourfolds have ℚ -rational points under a hypothesis on the Brauer–Manin obstruction. [ABSTRACT FROM AUTHOR]